LOCALLY DISCRETE EXPANDING GROUPS of ANALYTIC DIFFEOMORPHISMS of the CIRCLE Bertrand Deroin

LOCALLY DISCRETE EXPANDING GROUPS OF ANALYTIC DIFFEOMORPHISMS OF THE CIRCLE Bertrand Deroin

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Bertrand Deroin. LOCALLY DISCRETE EXPANDING GROUPS OF ANALYTIC DIFFEOMOR- PHISMS OF THE CIRCLE. Journal of topology, Oxford University Press, 2020. hal-03060432

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BERTRAND DEROIN

Abstract. We show that a ﬁnitely generated subgroup of Diﬀω(S1) that is expanding and locally discrete in the analytic category is analytically conjugated to a uniform lattice k 1 in PGL] 2 (R) acting on the k-th covering of RP for a certain integer k > 0.

1. Introduction In the study of the dynamics of finitely generated groups acting by analytic diffeomorphisms on the circle (or more generally in analytic unidimensional dynamics) the dichotomy discreteness versus non discreteness is very useful and important. Many interesting dy- namical properties can be easily established in the non locally discrete regime, for instance concerning ergodicity or minimality of the action. However in the locally discrete regime things are not completely understood yet, even if a conjectural classification is expected, see e.g. the survey [3]. The goal of this note is to provide such a classification under the additional assumption that the action is expansive, as announced in [3]. Expansive means that for every point of the circle there exists an element of the group whose derivative at that point is greater than 1 in modulus. Our main result (Corollary 10.2) shows that up to analytic conjugacy, only cocompact lattices of the finite cyclic coverings of PGL2(R) acting on the corresponding finite cyclic covering of the real projective line RP 1 are at the same time expansive and locally discrete in the analytic category. The precise definitions of expansiveness and local discreteness in the analytic category are exposed in sections 4 and 6 respectively. This result is part of a more general result concerning the dynamics of pseudo-groups of holomorphic maps on Riemann surfaces having both local discreteness and hyperbolicity properties. However, its proof in the particular case of the circle group action is consid- erably simpler (essentially because of the use of a combination of the convergence group theorem by Gabai [4] and Casson-Jungreis [2], and of the differentiable rigidity theory of Fuchsian groups by Ghys [5]), and deserves a special interest for the theory of circle group actions.

Organisation of the paper. Section 3 is devoted to review some aspects of the theory of hyperbolic groups that will be needed in our argument. In sections 4 and 5 (resp. 6) we present the deﬁnition of expansiveness (resp. local discreteness) that is assumed in our main result. Section 7 is devoted to the main technical tool of our method, namely the

Date: November 26, 2018. 1 2 BERTRAND DEROIN convergence property of the lines of expansion. The last three sections 8, 9 and 10 are devoted to the proof of our main result: the Corollary 10.2.

Acknowledgments – I express my gratitude to the team [1] who encouraged me to write this note. I particularly thank Michele Triestino for his careful reading.

2. Index of notations We denote by greek letters the distances in the Cayley graphs we consider. • S1 = R/Z • x, y, . . . points of S1 • B(x, r) : ball of radius r centered at x in C/Z for the euclidean metric • G subgroup of Diffω(S1) • S symmetric set of generators of G • kgk minimal number of generators needed to write g • δ constant of hyperbolicity for the Cayley graph associated to the pair (G, S) • f, g, . . . elements of G • s element of S x •{Em}m≥0 line of expansion at x x • xm = Em(x) • Dg derivative of the map g • Dx : G × G → R derivative cocycle • κ(g, E) distortion of the map g on the set E • a, b constant appearing in Lemma 6.2 (controlling size of derivatives) • c > 0 constant of uniform expansion of derivatives along the set S • c > 0 constant controlling logarithms of derivatives of elements of S, • γi’s constant of quasi-isometry appearing in Proposition 7.1 • Ω: S1 → ∂G equivariant map defined by equation (22) • φ : ∂G → RP 1 map that is ρ-equivariant • Φ = φ ◦ Ω ^k • for every positive integer k, PGL2(R) is the k : 1 cyclic covering of PGL2(R), acting k on the k : 1 cyclic covering R]P 1 of RP 1 • U ⊂ S1 × RP 1 complement of the graph of Φ • K ⊂ U × R compact subset defined by equation (24) • (x, z, t) coordinates of a point in U × R • ∆ ⊂ U × R fundamental domain for the G-action on U defined by equation (23) • M quotient of U by G ∂ • V = ∂t vector fields on M •F ±: weak stable foliations of V on M • D developing map LOCALLY DISCRETE EXPANDING GROUPS 3

3. Preliminaries of geometric group theory We review some notions of geometric group theory that will be useful for our argument. Let G be a finitely generated group. Given a finite symmetric generating subset S ⊂ G, we associate the norm kgk of an element g ∈ G as being the minimum number of elements of S that is needed to write g. The Cayley graph of the pair (G, S) is the non oriented graph whose vertices are the elements of G and the edges are the pairs {g, sg} with g ∈ G and s ∈ S. The group G acts naturally on its Cayley graph by right multiplications. The combinatorial distance d associated to this graph – namely, the one defined as the minimum number of edges one has −1 to cross to go from a vertex to another one – is given by the formula d(g1, g2) := g2g1 . Another finite symmetric generating set gives rise to another graph whose set of vertices is G, for which the identity map is bi-Lipschitz with respect to the associated distances. The group G is called hyperbolic if its triangles are thin, in the sense that there exists a constant δ > 0 such that for any triple of points g1, g2, g3 ∈ G, and any collections of geodesics [g1, g2], [g2, g3] and [g3, g1] between these points, we have that

δ δ (1) [g1, g3] ⊂ [g1, g2] ∪ [g2, g3]

where Aδ is the set of points at a distance from a point of A less than δ. A triangle in a graph satisfying an inequality such as (1) is called δ-thin. A geodesic ray parametrized by an interval I ⊂ Z is a sequence {gn}n∈I of elements of G such that d(gk, gl) = |k − l| for every k, l ∈ I. The set of equivalence classes of geodesic rays parametrized by N up to bounded Hausdorﬀ distance, is the geometric boundary of G, and is denoted by ∂G. It is equipped with the quotient of the topology of simple convergence. In the case where G is Gromov hyperbolic, this topological space is a compact metric space. Moreover, in this latter case, the group G acts naturally on its boundary by homeomorphisms, and the action is minimal unless G is virtually cyclic. Given constants α ≥ 1 and β ≥ 0, an (α, β)-quasi-geodesic on G is a sequence {gn}n∈N of elements of G such that for every m, n ∈ N, we have

−1 (2) α |n − m| − β ≤ d(gm, gn) ≤ α|n − m| + β.

We recall that there exists a constant η such that two (α, β)-quasi-geodesics having the same extremities are bounded appart in Hausdorﬀ distance by less than η. As a consequence of this, up to enlarging the constant δ, any (a, b)-quasi-geodesic triangle is δ-thin. The Gromov product is deﬁned by d(g, h ) + d(g, h ) − d(h , h ) (3) (h , h ) := 1 2 1 2 , 1 2 g 2

for every g, h1, h2 ∈ G. Its geometrical signiﬁcance in a Gromov hyperbolic graph is that, up to some constant depending only on the constants of hyperbolicity of the graph, the Gromov product (h1, h2)g is the distance from g to a (any) geodesic between h1 and h2. Given two points p, q ∈ ∂G, this Gromov product can be extended to points of the 4 BERTRAND DEROIN

boundary of G by the following formula

(4) (p, q)g := sup lim sup (hm, kn)g, (hm)m, (kn)n m,n→∞

where the first supremum is taken over all geodesics {hm}m∈N and {kn}n∈N that tend to p and q respectively. In fact, if (hm)m and (kn)n are geodesics tending to p and q respectively, then if m, n are sufficiently large, the quantity (hm, kn)g differs from (p, q)g by an additive term which is bounded by a constant that depends only on the hyperbolicity constants of G. Then two points p, q ∈ ∂G are close to each other if and only if the Gromov product (p, q)e is large.

4. Expanding property and derivative cocycle Definition 4.1. A subgroup G ⊂ Diffω(S1) is expanding if for every point x ∈ S1, there exists an element g ∈ G such that log |Dg(x)| > 0. Remark 4.2. By compacity of S1, if G ⊂ Diffω(S1) is a subgroup which is expanding, there exists a finite subset S ⊂ G and a constant c > 0 such that for every point x ∈ S1 there exists s ∈ S such that log |Ds(x)| ≥ c. If G is finitely generated, we can assume furthermore that S is a symmetric generating set. Definition 4.3. If G is a subgroup of Diffω(S1), for every x ∈ S1, we define the derivative cocycle Dx by the formula −1 (5) Dx(g1, g2) := log |Dg2(x)| − log |Dg1(x)| = log |D(g2 ◦ g1 )(g1(x))|. Remark 4.4. The derivative cocyle satisfies the obvious equivariance −1 −1 (6) Dg(x)(g1 ◦ g , g2 ◦ g ) = Dx(g1, g2). 1 for every x ∈ S , and g, g1, g2 ∈ G. Moreover, if G is generated by a finite symmetric subset S, the derivative cocycle satisfies

(7) Dx(g1, g2) ≤ c d(g1, g2), where (8) c := max log |Ds(x)|. x∈S1,s∈S Definition 4.5. Let G ⊂ Diffω(S1) be a finitely generated subgroup, S ⊂ G a finite symmetric generating subset, c > 0, and x ∈ S1.A c-line of expansion relative to the x point x is a sequence {En}n≥0 of elements of G such that for every n > 0 one can write x x x x En = snEn−1 for some sn ∈ S, and for every 0 ≤ m ≤ n we have x x (9) Dx(Em,En) ≥ c(n − m). Remark 4.6. Observe that if S and c are given by Remark 4.2, for any 0 < c ≤ c, for any point x ∈ S1, and for every g ∈ G, there exists a c-line of expansion relative to the point x starting at g. In the sequel, we will choose either c = c or c = c/2. LOCALLY DISCRETE EXPANDING GROUPS 5

Lemma 4.7. Given a finitely generated group G ⊂ Diffω(S1), a finite symmetric generating subset S ⊂ G, and a constant c > 0, there exists α ≥ 1 such that any c-line of expansion is a (α, 0)-quasi-geodesic.

x x Proof. We clearly have d(Em,En) ≤ |m − n| for every m, n so the right hand side of (2) is satisfied with α = 1 and β = 0. Moreover, for any m, n ∈ N with m ≤ n, (7), (8) together with (9) show that x x x x c(n − m) ≤ Dx(Em,En) ≤ c d(Em,En) so that the left hand side of (2) is satisfied with α = c/c and β = 0. 5. Bounded distortion along lines of expansion Definition 5.1. The distortion of a differentiable analytic (resp. holomorphic) map g : U → V between open subsets of S1 (resp. C ) in restriction to a subset E ⊂ U is the quantity |Dg(y)| (10) κ(g, E) := max log . x,y∈E |Dg(x)| Lemma 5.2. Let G be a subgroup of Diffω(S1) generated by a finite symmetric set S, and let c > 0 be some constant. There exists r > 0 such that the following holds. For any x 1 c-line of expansion (En)n≥0 relative to some x ∈ S , and for every n ∈ N, the element x −1 x −1 (En) ∈ G has a holomorphic univalent extension (Ên) to the ball B(xn, r) of center x xn = En(x). Proof. Choose a real number r > 0 small enough so that the following condition holds: every s ∈ S extends as a univalent holomorphic map se defined on the r-neighborhood Ar of S1 in C/Z, and moreover for every y ∈ S1

(11) κ(s,e B(y, r)) ≤ c. x Let 0 ≤ m ≤ n be an integer. Since (En)n≥0 is a c-line of expansion at x, we have that x −1 −c |D(sm) (xm)| ≤ e . So (11) shows that for every y ∈ B(xm, r)

x −1 c x −1 |D(s^m) (y)| ≤ e · |D(sm) (xm)| ≤ 1.

x −1 Hence, the ball B(xm, r) is sent by (^sm) inside the ball B(xm−1, r). x −1 x −1 x −1 Since (En) := (s1) ... (sn) , it has a holomorphic extension to B(xn, r), which maps this latter to B(x, r). To end this paragraph, recall the following distortion estimate, due to Koebe [8]:

Lemma 5.3 (Koebe). There is a constant κ > 0 such that the following holds. Let z1, z2 be points of C, r > 0 be a positive real number, and g be a univalent non constant holomorphic 0 map from B(z1, r) to C sending z1 to z2. Then for every 0 ≤ r ≤ r/2, 0 κ(g, B(z1, r )) ≤ κ. 0 0 −κ Moreover, g(B(z1, r )) contains the ball B(z2, r e |Dg(z1)|). 6 BERTRAND DEROIN

6. Local discreteness For the next definition, we think of the circle as being the one dimensional submanifold S1 = R/Z contained in the Riemann surface C/Z. Definition 6.1. A subgroup G ⊂ Diffω(S1) is non locally discrete in the analytic category at a point x ∈ S1, if there exists a neighborhood V ⊂ C/Z of x and a sequence {gn}n≥1 of elements of G\{id} that extend as univalent holomorphic maps from V to C/Z and whose extensions to V tend uniformly to the identity on V when n tends to infinity. Otherwise, G is locally discrete in the analytic category at the point x. We have the following simple characterization of the local discreteness of G everywhere in the analytic category. For every x ∈ C/Z, and every non negative real number r, we denote by B(x, r) the ball of radius r centered at x in C/Z, for the euclidean distance. Lemma 6.2. Let G ⊂ Diffω(S1) be a finitely generated subgroup, which is locally discrete in the analytic category at every point. Let S be a finite symmetric generating set for G. Then, given numbers r > 0 and a < b, there is an integer γ ∈ N∗ such that the following holds. Let y ∈ S1 and f ∈ G. Suppose that f can be extended as a univalent holomorphic map fe defined on the ball B(y, r) that satisfies a ≤ log |Dfe| ≤ b on B(y, r). Then f is a composition of at most γ elements of S, namely kfk ≤ γ.

Proof. By contradiction, suppose that this is not true. Then there is a sequence (fk)k 1 of elements of G and a sequence (yk)k of points of S such that for every k, fk has a holomorphic univalent extension fek defined on B(yk, r) whose derivative on this latter satisfies a ≤ log |Dfek| ≤ b, and whose word-length tends to infinity. Taking a subsequence 1 if necessary, we can suppose that (yk)k converges to a point y ∈ S when k tends to infinity. Then, the maps fek are defined on the ball B(y, r/2) when k is large enough. Since logarithm of derivatives are bounded on B(y, r/2), Montel’s theorem shows that, taking a

subsequence if necessary, the maps fek converge uniformly to a holomorphic univalent map −1 fe : B(y, r/4) → C/Z. In particular, the maps fgk+1 ◦ fek are well-defined on B(y, r/8) if k is large enough, and converge to the identity in the uniform topology when k tends to infinity. The discreteness assumption implies that for k large enough, fk+1 = fk; this contradicts the fact that the word-length of fk tends to infinity. The following result shows that for an expanding group of analytic diffeomorphisms of the circle, being locally discrete in the analytic category somewhere is the same notion as being locally discrete in the analytic category everywhere. Lemma 6.3. If a subgroup G ⊂ Diffω(S1) is expanding, then it is either non locally discrete in the analytic category everywhere, or locally discrete in the analytic category everywhere. Proof. The set of points where the group is non locally discrete in the analytic category is obviously an open set. Its complement, the set where the group is locally discrete in the analytic category, is a closed invariant subset Λ.ˆ Suppose that Λˆ is non empty. LOCALLY DISCRETE EXPANDING GROUPS 7

We claim that Λˆ does not contain a finite orbit in view of the expansiveness assumption. Indeed, any point whose orbit is finite is a hyperbolic fixed point of some element of G which takes the form f : z 7→ λz with |λ| < 1 in some linearizing coordinates. Since the group is not virtually abelian (by expansiveness) the stabilizer of z = 0 contains an element tangent to the identity at z = 0 of the form g : z 7→ z + azk + ... with a 6= 0 and k ≥ 2. Now, we have for a positive integer n (12) f −n ◦ g ◦ f n = z + aλ(k−1)nzk + ... which shows that there exists a neighborhood of z = 0 in C where f −n ◦ g ◦ f n converges to the identity when n tends to +∞. This shows that the finite orbits are contained in the non locally discrete part. Hence Λˆ contains an exceptional minimal subset Λ. However, a result of Hector [7] shows that the stabilizers of the components of the complement of Λ are virtually cyclic. Hence every point in the complement of the exceptional minimal set is locally discrete in the analytic category (in fact in the compact open topology). In particular, the group is locally discrete everywhere, and the result follows. In view of this result, we will call an expanding group locally discrete in the analytic category if it is locally discrete in the analytic category at some point, and hence at every point.

7. Convergence of lines of expansion in the Cayley graph Proposition 7.1. Let G ⊂ Diffω(S1) be a finitely generated subgroup which is expanding and locally discrete in the analytic category. Let S be a finite symmetric subset of generators of G, c := sup log |Ds(x)|, x∈S1,s∈S

and let c > 0 be some constant. Then, there are constants γ1, γ2, γ3 > 0 such that the 1 x x following holds. Let x ∈ S , and (Em)m, (Fn )n be c-lines of expansion relative to the point x x x. Then for every non negative integers n, m ≥ γ1d(E0 ,F0 ) + γ2, such that x x (13) Dx(Em,Fn ) ≤ c, x x we have d(Em,Fn ) ≤ γ3. x x Proof. Moving x to E0 (x) if necessary, we can assume that E0 = e (see Remark 4.4). We x y x −1 y will then write g := F0 , y = g(x) and En := Fn ◦ g . The sequence {En}n≥0 is then a c-line of expansion relative to the point y, which explains the notation. Observe then that x x x x y kgk = d(E0 ,F0 ). We denote xm := Em(x) and yn := Fn (x) = En(y). x −1 The map (E^m) is deﬁned on B(xm, r), where r is the constant given by Lemma 5.2, and Koebe’s Lemma shows x −1 κ (E^m) ,B(xm, r/2) ≤ κ, 8 BERTRAND DEROIN

and eκr ^x −1 (Em) (B(xm, r/2)) ⊂ B x, x . 2|DEm(x)| Since the map g is a composition of kgk elements of S, it has a univalent holomorphic −(kgk−1)c extension ge defined on the ball B(x, re ), that takes values in the ball B(y, r). Koebe’s Lemma shows −(kgk−1)c κ g,e B x, re /2 ≤ κ. ^x −1 In order that the composition ge◦(Em) be defined on B(xm, r/2) with distortion bounded by 2κ, a sufficient condition is then eκr re−(kgk−1)c (14) x ≤ , 2|DEm(x)| 2 or equivalently x (15) log |DEm(x)| ≥ (kgk − 1)c + κ. In this case we have for every r0 ≤ r/2 ^x −1 0 2κ 0 x −1 ge ◦ (Em) (B(xm, r )) ⊂ B(y, e r |D(g ◦ (Em) )(xm)|), x because (Em)m is a c-line of expansion. Observe that condition (15) is fulfilled if κ − c (16) m ≥ γ kgk + γ where γ = c/c, γ = . 1 2 1 2 c

y −1 Similarly, the map (Ên) is defined on the ball of radius r, and we have −κ e r y −1 B y, y ⊂ (Ên) (B(yn, r/2)) 2|DEn(y)| and y −1 κ (Ên) ,B(yn, r/2) ≤ κ.

y e−κr Then the extension Efn is well-defined and univalent on the ball B y, y and its 2|DEn(y)| distortion is bounded by −κ y e r κ Efn,B y, y ≤ κ. 2|DEn(y)| In particular, for every r0 ≤ r/2, we have, under the condition −κ 2κ 0 x −1 e r (17) e r |D(g ◦ (Em) )(xm)| ≤ y , 2|DEn(y)| or equivalently x y 0 (18) 3κ + Dx(Em,En ◦ g) ≤ log(r/2r ), y ^x −1 0 that the composition Efn ◦ ge ◦ (Em) is defined on the ball B(xm, r ) and its distortion is bounded by 3κ. Condition (18) is satisfied if log(r/2r0) ≥ 3κ + c. LOCALLY DISCRETE EXPANDING GROUPS 9

0 With this choice of r > 0, we can then apply Lemma 6.2 with γ3 = γ to get the conclusion.

8. Gromov hyperbolicity of G Proposition 8.1. Let G ⊂ Diﬀω(S1) be a ﬁnitely generated subgroup which is expanding and locally discrete in the analytic category. Then, G is Gromov hyperbolic. Proof. We will use the following result [9, Theorem 2.11] of Nekrashevych. Fix x ∈ S1, and denote by Γx the directed graph whose vertices are the elements of 2 G, and whose directed edges are the couples (g0, g1) ∈ G such that

(19) Dx(g0, g1) ≥ c/2 and d(g0, g1) ≤ 2, where c > 0 is the constant given by Remark 4.2. The set G is equipped with the combi- x natorial metric dΓx induced by the underlying undirected graph induced by Γ : dΓx (g1, g2) x is the minimum number of undirected edges of Γ necessary to go from g1 to g2. Hence, the combination of Proposition 7.1 and of [9, Theorem 1.2.9] with the following constants x η = c/4, ∆ = c, show that Γ equipped with its distance dΓx is Gromov hyperbolic.

Claim: The inclusion (G, dΓx ) → (G, d) is a quasi-isometry

Proof. From the deﬁnition of Γx, we immediately have

(20) d ≤ 2dΓx .

Let {g1, g2} be an edge of G, i.e. g2 ∈ Sg1. We can assume that Dx(g1, g2) ≥ 0 up to x exchanging g1 and g2. If Dx(g1, g2) ≥ c/2, then the directed arrow g1 → g2 belongs to Γ . x If not, let s ∈ S be an element such that Dx(g1, sg1) ≥ c given by Remark 4.2. Then Γ contains the directed edges: g1 → sg1 (by deﬁnition of s) and g2 → sg1. So the dΓx -distance between g1 and g2 in the graph is bounded by 2. In particular, we get

(21) dΓx ≤ 2d. The claim follows from equations (20) and (21). The Proposition follows from the claim and the fact that Gromov hyperbolicity is a quasi-isometric invariant.

9. The group G is virtually a Fuchsian group A consequence of Proposition 8.1 is that we have a well-defined map (22) Ω : S1 → ∂G which associates to a point x ∈ S1 the equivalence class in ∂G of a c-line of expansion at the point x (here c is the constant appearing in Remark 4.2). Indeed, Proposition 7.1 shows that two such lines of expansion are at a bounded Hausdorff distance from each other. Proposition 9.1. The map Ω: S1 → ∂G is a finite covering. 10 BERTRAND DEROIN

Proof. The proof is organized as a sequence of claims. Claim 0 – The map Ω: S1 → ∂G is equivariant and continuous. Proof. The equivariance is immediate. Let us prove the continuity at a point x ∈ S1. x Suppose {Em}m is a c-line of expansion at x where m > 0 is the constant given by Remark 1 4.2. Let m0 be a large integer. For y ∈ S in a sufficiently small neighborhood of x, x x x x denoting ym := Em(y), and recalling that Em = sm ◦ Em−1 (see Definition 4.5), we have x log |Dsm(y)| ≥ c/2 for every 0 ≤ m ≤ m0. y Hence, one can define a c/2-line of expansion {Em}m≥0 relative to y by y x Em := Em if m ≤ m0 y and by taking for {Em}m≥m0 a c-line of expansion relative to y. The two c/2-lines of x y expansion {Em}m and {Em}m relative to x and y respectively coincide for m ≤ m0 and converge respectively to ωx and ωy. Hence, being (α, 0)-quasi-geodesics for a constant α depending only on c and S (see Lemma 4.7) their limit points ωx and ωy are close to each other in ∂G. This proves continuity of Ω. Claim 1 – There are constants c, d > 0 such that, for every x ∈ S1, any geodesic ray {gn}n≥0 in G starting at g0 = e and tending to ω ∈ ∂G satisfies

Dx(e, gn) ≥ cn − d for n ≤ (Ω(x), ω)e and Dx(e, gn) ≤ −cn + c · (Ω(x), ω)e + d for n ≥ (Ω(x), ω)e.

x x x Proof. Let {El }l≥0 and {Fm}m≥0 be c-lines of expansion relative to x beginning at E0 = e x and F0 = gn respectively. By Proposition 7.1, there exist integers L and M such that x x d(EL,FM ) ≤ γ3.

Let α be given by Lemma 4.7, and let β = γ3. The (α, β)-quasi-geodesic triangle formed by the segments x x {gs}0≤s≤n, {El }0≤l≤L and {Fm}0≤m≤M , is δ-thin for a certain constant δ depending only on (α, β) and the hyperbolicity constants of (G, d). So for some integer N, the segments {gs}0≤s≤(Ω(x),ω)e and {gs}n≥s≥(Ω(x),ω)e are δ-close to c-segments of expansion relative to x. The conclusion follows. Claim 2 – There is a number M ∈ N∗ such that the level subsets Ω−1(ω), for ω ∈ ∂G, have cardinality less than M. k −1 Proof. Indeed, let x ∈ Ω (ω), k = 1,...,M, be distinct points, and let {gn}n be a geodesic ray from e to ω ∈ ∂G. For every k = 1,...,M, the sequence {gn} is O(δ)-close xk k xk to a c-line of expansion {Em }m≥0 relative to x and starting at E0 = e. Let n be a large integer. For every k = 1,...,M, there exists m such that d(g ,Exk ) = k n mk O(δ). Let r > 0 be the constant given by Lemma 5.2. Denoting xk = Exk (xk) for n ≥ 0 n mk LOCALLY DISCRETE EXPANDING GROUPS 11

and k = 1,...,M, Lemma 5.2 shows that the map (E^xk )−1 extends as a univalent map mk k k reκ deﬁned on B(x , r/2) whose image lies in the ball B(x , k ). In particular, there n 2|Dgn(x )| 0 k k −1 exists r > 0 depending only on r and δ such that, denoting yn = gn(x ), the map gn k 0 k reκ sends the interval B 1 (y , r ) inside the interval B 1 (x , k ). S n S 2|Dgn(x )| k reκ For n large enough, the intervals B 1 (x , k ), k = 1,...,M, are disjoint, hence so S 2|Dgn(x )| k 0 0 are the intervals BS1 (xn, r )’s. In particular, M ≤ 1/r , and the claim follows. Let K be the set of compact subsets of S1 equipped with the topology induced by the Hausdorﬀ distance. Claim 3 – The map ω ∈ ∂G 7→ Ω−1(ω) ∈ K is continuous.

k ∞ Proof. Let {ω }k be a sequence of points of ∂G tending to ω ∈ ∂G. For every k ∈ N, k k let {gn}n≥0 be a geodesic ray tending to ω . Up to extracting if necessary, one can assume k k ∞ that for each n, the sequence {gn}k≥0 is stationary, namely for k ≥ k(n), gn = gn . The ∞ ∞ sequence {gn }n is a geodesic ray tending to ω . Using the same notations as in Claim 1, −1 k k the Hausdorﬀ distance between Ω (ω ) and the set {log |Dgn| ≥ 0} is less than ε for every n ≥ n(ε), for every k ∈ N ∪ {∞}. Applying this to k ≥ k(n(ε)), we get that the Hausdorﬀ −1 k −1 ∞ distance between Ω (ω ) and Ω (ω ) is less than 2ε, which proves the claim.

Claim 4 – The function k : ω ∈ ∂G 7→ |Ω−1(ω)| ∈ N is constant. Proof. Notice that G cannot be virtually cyclic since otherwise its action on the circle could not be expanding. By [6, Chapitre 8], it acts minimally on its boundary. Claim 3 shows that k is lower semi-continuous, and it is G-invariant. In particular, the subset {k = min k} ⊂ ∂G is closed, non empty, and G-invariant. The action of G on ∂G being minimal, it must be the whole ∂G, hence the conclusion holds. Claims 2, 3 and 4 show that Ω is a covering. Corollary 9.2. The action of G on S1 is topologically conjugated to the action of a (co- k ^k 1 compact) lattice of PGL2(R) on the k-th covering R]P for a certain integer k > 0. Proof. By Proposition 9.1, the action of G on S1 is topologically conjugate to a ﬁnite covering of its action on its boundary. This proves that the boundary of G is homeomorphic to the circle, hence the result follows from the convergence group theorem [4, 2]. The lattice is cocompact since otherwise the group G would be virtually free, and its boundary would be a Cantor set.

10. Differentiable rigidity It is presumably well-known that Corollary 9.2, together with the diﬀerentiable rigidity theory developed by Ghys in [5], imply our main result, the Corollary 10.2. However, this implication is not directly stated in this form in the litterature, so we provide a detailed proof below. 12 BERTRAND DEROIN

Proposition 10.1. Let G ⊂ Diffω(S1) be a finitely generated subgroup which is expanding and locally discrete in the analytic category. Then G preserves an analytic RP 1-structure. Proof. We first observe that we can assume that the group G preserves the orientation on S1. Indeed, suppose that we know that the subgroup G+ of elements of G that preserves the orientation on S1 preserves an analytic projective structure σ on S1. If G+ = G we are done. If not, let g be an element of G that reverses the orientation. Since G is generated by G+ and g, it suffices to prove that g preserves the projective structure σ as well. For this, let us consider the quadratic differential defined by S(g) := {g, z}dz2, where {g, z} 2 D3g 3 D2g is the Schwarzian derivative {g, z} = Dg − 2 Dg computed in projective coordinates of σ. The cocycle relations satisfied by the Schwarzian derivative shows that S(g) is a well-defined quadratic differential on the circle, namely it is independant of the chosen projective coordinates. We need to prove that S(g) = 0 identically. For every f ∈ G+, there exists h ∈ G+ such that g◦f = h◦g. The cocycle relation satisfied by the Schwarzian derivative, together with the fact that both f and h preserve σ, implies S(g) = f ∗S(g), hence proving that S(g) is invariant by the whole group G+. In particular, if S(g) does not vanish identically, the volume p|S(g)| is G+-invariant, which contradicts the fact that G+ is expanding on the support of S(g). So from now on we will assume that G preserves orientation on S1. Denote by φ : ∂G → RP 1 a homeomorphism that conjugates the G-action on its boundary to the action of a 1 Fuchsian group Γ ⊂ PGL2(R) on RP (see Corollary 9.2 for the existence of φ). We can assume that φ preserves orientation, so that the lattice Γ is indeed contained in PSL2(R). We denote by ρ : G → Γ the representation that satisfies φ(gω) = ρ(g)φ(ω) for every g ∈ G, ω ∈ ∂G. Let U ⊂ S1 × RP 1 be the complement of the graph of Φ = φ ◦ Ω, namely the set of points (x, z) ∈ S1 × RP 1 such that z 6= Φ(x). Consider the analytic action of G on U × R defined by: (23) g(x, z, t) = (g(x), ρ(g)(z), t + log |Dg(x)|).

Claim 0 – The action (23) is free, proper discontinuous and cocompact. Hence the quotient of U by G is a closed analytic 3-manifold M. Freeness. Assume that there exists a point (x, z, t) which is fixed by an element g ∈ G. The theory of hyperbolic groups (see e.g. [6, Chapitre 8, §3]) tells us that g is either n of finite order, or a hyperbolic element, in which case the sequence {g }n∈N is a quasi- geodesic. Assume by contradiction that g is hyperbolic. It has a fixed point x on S1 having a derivative equal to 1 at x – recall that we assume the orientation is preserved by n G – so all the numbers Dx(e, g ) are zero. This contradicts the claim 1 of the proof of n Proposition 9.1 since the sequence {g }n≥0 is a quasi-geodesic. This contradiction shows that g has finite order in G. Its image ρ(g) is an element of PSL2(R) of finite order that must be the identity since it fixes the point z ∈ RP 1. Hence, g lies in the kernel of ρ which LOCALLY DISCRETE EXPANDING GROUPS 13

is a cyclic subgroup of G acting freely on S1. Since g ﬁxes the point x, it is therefore the identity map. Hence the action is free as claimed. Proper discontinuity. Any compact set in U is contained in some compact set of the sort

−1 (24) K := {(x, z, t) ∈ U × R | (Ω(x), φ (z))e ≤ C and |t| ≤ T }, for some constants C,T . We must prove that there is only a finite number of elements g of G such that gK ∩ K 6= ∅. Let g ∈ G, and suppose that for some (x, z, t) ∈ K we have −1 g(x, z, t) ∈ K. Since both (Ω(x), z)e and (gΩ(x), gφ (z))e are bounded by C, g must lie at a distance from a geodesic between φ−1(z) and Ω(x) bounded by some constant depending only on C and the constants of hyperbolicity of (G, d). But a geodesic from φ−1(z) to Ω(x) lies at a finite distant from a c-line of expansion between φ−1(z) and Ω(x) relative to x, the Hausdorff distance being bounded by some constant depending only on c and G, S. Hence, we have | log |Dg(x)|| ≥ cst kgk + cst for some positive constants depending only on c and G. However | log |Dg(x)|| ≤ 2T since both (x, z, t) and g(x, z, t) lie in K. This proves that the norm of g is bounded, hence there is only a finite number of g ∈ G such that gK ∩ K 6= ∅, and the properness of the action of G on U × R follows. Cocompactness. The cocompactness of the action of G on U × R follows from cohomo- logical reasons, but it is instructive to prove it by hands. We will prove that any point (x, z, t) ∈ U × R can be sent by an element of G to a point (x00, z00, t00) belonging to some compact set defined by equation (24), for some constants C,T that depend only on G. Let us first find g ∈ G such that ρ(g)(Φ(x)) and ρ(g)(z) are separated by a constant that depends only on G. Let {gn}n≥0 be a geodesic ray on G tending to Ω(x). A consequence of Lemma 5.2 is that, if n is large enough so that |Dgn(x)| is larger than the inverse of −1 −1 the distance between x and Φ (z), the distance between gn(x) and the set Φ (ρ(gn)z) is larger than some constant depending only on G. In particular, the points Φ(gn(x)) and ρ(gn)(z) are separated by a constant that depends only on G. Similarly, suppose that a −1 geodesic ray {gn}n≥0 tends to φ (z) when n tends to infinity. Then, the sequence {gn}n≥0 lies at a finite distant from a line of expansion at z (for the ρ-action of G on RP 1 given by ρ, which is locally discrete and expansive), and the same argument applies: namely, for n large enough, the distance between ρ(gn)(Ω(x)) and ρ(gn)(z) is bounded from below by some constant depending only on G. So we are done. Now write (x0, z0, t0) = g(x, z, t), with Ω(x0) and z0 separated by a constant depending 0 00 00 00 00 x0 0 0 0 only on G. If t ≤ 0, then the t -variable of the point (x , z , t ) := En (x , z , t ) grows linearly with n, while keeping the distance between Φ(x00) and z00 separated by a constant depending only on G. Hence for some n the point (x00, z00, t00) belongs to the compact set defined in equation (24) for some constants C,T depending only on G. If t00 > 0, let −1 0 {gn}n≥0 be a geodesic ray beginning at g0 = e and tending to φ (z ) when n tends to ∞. Because Φ(x0) and z0 are separated by some constant depending only on G, the Gromov 0 −1 0 product (Ω(x ), φ (z ))e is bounded by some constant depending only on G as well, and 0 the claim 1 of the proof of the Proposition 9.1 shows that log |Dgn(x )| decreases linearly 00 00 00 0 0 0 when n tends to ∞. Letting (x , z , t ) := gn(x , z , t ), and reasoning as before, we infer 00 −1 00 that the Gromov product (Ω(x ), φ (z ))e remains bounded by some constant depending 14 BERTRAND DEROIN

only on G, and for some n, the t00-coordinates enters in some fixed interval of the form [−T,T ] for some constant T depending only on G as well. The conclusion follows. ∂ Claim 1 – The flow on M induced by the non singular vector fields V := ∂t is Anosov, and its weak stable foliation F + is given by the equation dx = 0. Proof. Let ∆ ⊂ U × R be a fundamental domain for the G-action given by (23). Let (x, z, t) ∈ ∆ be some point, and s be a negative real number. There exists an element gs of G such that gs(x, z, t) ∈ ∆. This element gs can be chosen as the n-th term of an expansive line at the point x beginning at e, with n growing linearly with −s and constants depending only on G. Hence log |Dgs(x)| (resp. log |Dρ(gs)(z)|) increases linearly to +∞ with −s (resp. decreases linearly to −∞ with −s). As a consequence: when s tends to +∞, the flow exp(sV ) contracts exponentially a metric on the bundle T F −/RV , where F − is the foliation defined on the covering U by z = cst, whereas it expands exponentially a metric on the foliation F ++ defined on the covering U by (x, t) = cst. The conclusion follows. The weak unstable foliation F + defined on the covering U by x = cst is analytic, hence a Theorem of Ghys [5, Théorème4.1] shows that F + has an analytic transverse projective structure. This latter lifts to a transverse projective structure on the foliation of U defined by the submersion x : U → S1, which is invariant by G, hence gives birth to an analytic projective structure on S1 which is invariant by G. The proof of Proposition 10.1 is complete. Corollary 10.2. A finitely generated subgroup of Diffω(S1) which is expanding and locally k discrete in the analytic category is analytically conjugated to a uniform lattice in PGL]2(R) acting on the k-th covering of RP 1 for a certain integer k > 0. Proof. By Proposition 10.1, the group G preserves an analytic RP 1-structure on S1. Let D : R → RP 1 be a developing map for this structure: this is an analytic local diffeomor- phism which is equivariant with respect to an element A ∈ PGL2(R), namely we have (25) D(˜x + 1) = AD(˜x) for everyx ˜ ∈ R. The group G lifts to a subgroup Ge ⊂ Diffω(R) that commutes with the translationx ˜ 7→ x˜ + 1. Since the RP 1-structure is invariant by G, there exists a representation ρ˜0 : Ge → PGL2(R) which is such that (26) D ◦ g˜ = ρ˜0(˜g) ◦ D. We then have A = ρ˜0(˜x 7→ x˜ + 1). Hence A commutes with ρ˜0 Ge . We will prove that A = Id. Assume by contradiction that A has no fixed point on RP 1. In this case Ge is 1 contained in a conjugate of PO2(R), and in particular preserves a measure on RP with an analytic density. Its preimage by D is invariant by Ge, in particular by the translation x˜ 7→ x˜ + 1, hence produces on S1 a measure invariant by G that has an analytic density as well. This contradicts the expanding property for G. Assume by contradiction that A LOCALLY DISCRETE EXPANDING GROUPS 15

has some fixed point but is not the identity. Then D−1(Fix(A)) is a discrete Ge-invariant subset of R, and projects in S1 to a G-invariant finite orbit. This is a contradiction since the image of this latter by the map Ω would be a finite G-orbit in ∂G. The only remaining possibility is that A = Id as claimed. 0 0 In particular, the representation ρ˜ induces a representation ρ : G → PGL2(R), and (25) shows that D induces a finite covering from S1 to RP 1 which is ρ0-equivariant. Since G is a finitely generated, locally discrete and expanding subgroup of Diffω(S1), the same 0 is true for the image of ρ , hence this latter is a uniform lattice in PGL2(R). The result follows. References [1] S. Alvarez, D. Filimonov, V. Kleptsyn, D. Malicet, C. M. Coton,´ A. Navas, M. Triestino. Groups with infinitely many ends acting analytically on the circle. arXiv:1506.03839 [2] A. Casson, D. Jungreis. Convergence groups and Seifert fibered 3-manifolds. Inventiones Mathematicae 118 (1994), no. 3, 441–456. [3] B. Deroin, D. Filimonov, V. Kleptsyn, A. Navas. A paradigm for codimension one foliations. Geometry, dynamics, and foliations 2013, 59–69, Adv. Stud. Pure Math., 72, Math. Soc. Japan, Tokyo, 2017. [4] D. Gabai. Convergence groups are Fuchsian groups. Annals of Mathematics 136 (1992), no. 3, 447–510. [5] E.´ Ghys. Rigiditédifférentiable des groupes Fuchsiens. Publ. Math. IHES Tome 78 (1993), p.163-185. [6] E.´ Ghys & P. de la Harpe. Sur les groupes hyperboliques d’aprèsM. Gromov. Progress in Math. 83 1990 BirkhäuserBoston [7] G. Hector. Actions de groupes de difféomorphismesde [0, 1]. Géométriedifférentielle, Col- loque, Santiago de Compostela, 1972; Lecture Notes in Math. no 392, pp. 14–22. Springer, Berlin. [8] P. Koebe. Uber¨ die Uniformisierung der algebraischen Kurven. I. Math. Ann. 67 (1909), no. 2, p. 145-224. [9] V. Nekrashevych. Hyperbolic groupoids and duality. Memoirs of the AMS (2015)

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